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Consider a one dimensional Schrödinger operator $\tilde{A}=-\ddot{u}+V\cdot u$ with a periodic potential $V(\cdot )$, defined on a suitable subspace of $L^2(ℝ)$. Its spectrum is the union of closed intervals, and in general these intervals are separated by open intervals (spectral gaps). The Borg theorem states that we have no gaps if and only if the periodic potential $V(\cdot )$ is constant almost everywhere. In this paper we consider families of Finite Difference approximations of the operator $\tilde{A}$, which depend upon two parameters $n$, i.e., the number...